\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  -6.0 & +  7.00 x_{1} & +  7.00 x_{2} & +  1.00 x_{3} & +  3.00 x_{4} & -4.00 x_{5} & -3.00 x_{6} & -7.00 x_{7} & +  7.00 x_{8}\\
 x_{10}   &  26.0 & -5.00 x_{1} & -6.00 x_{2} & + 10.00 x_{3} & -5.00 x_{4} & -8.00 x_{5} & -1.00 x_{6} & -3.00 x_{7} & -1.00 x_{8}\\
 x_{11}   &  18.0 & -3.00 x_{1} & -8.00 x_{2} & -7.00 x_{3} & -1.00 x_{4} & -6.00 x_{5} & -4.00 x_{6} & +  3.00 x_{7} & +  5.00 x_{8}\\
 x_{12}   &  -5.0 & -5.00 x_{1} & +  8.00 x_{2} & -5.00 x_{3} & +  8.00 x_{4} & -3.00 x_{5} & +  6.00 x_{6} & + 10.00 x_{7} & +  2.00 x_{8}\\
 x_{13}   &  0.0 & +  4.00 x_{1} & -1.00 x_{2} & +  2.00 x_{3} & -5.00 x_{4} & +  5.00 x_{5} & -2.00 x_{6} & +  1.00 x_{7} & +  6.00 x_{8}\\
 x_{14}   &  1.0 & +  3.00 x_{1} & -10.00 x_{2} & +  2.00 x_{3} & +  3.00 x_{4} & +  8.00 x_{5} & +  5.00 x_{6} & +  1.00 x_{7} & -5.00 x_{8}\\
 x_{15}   &  -17.0 & -4.00 x_{1} & -9.00 x_{2} & +  8.00 x_{3} & +  5.00 x_{4} &   & -7.00 x_{6} & +  4.00 x_{7} & +  8.00 x_{8}\\
\hline
z    &  0.0 & +  1.00 x_{1} & +  5.00 x_{2} & -2.00 x_{3} & -4.00 x_{4} & +  4.00 x_{5} & +  5.00 x_{6} & +  3.00 x_{7} & +  4.00 x_{8}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{8}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,15$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -7.00 y_{9} & +  5.00 y_{10} & +  3.00 y_{11} & +  5.00 y_{12} & -4.00 y_{13} & -3.00 y_{14} & +  4.00 y_{15}\\
 y_{2}   &  1.0 & -7.00 y_{9} & +  6.00 y_{10} & +  8.00 y_{11} & -8.00 y_{12} & +  1.00 y_{13} & + 10.00 y_{14} & +  9.00 y_{15}\\
 y_{3}   &  1.0 & -1.00 y_{9} & -10.00 y_{10} & +  7.00 y_{11} & +  5.00 y_{12} & -2.00 y_{13} & -2.00 y_{14} & -8.00 y_{15}\\
 y_{4}   &  1.0 & -3.00 y_{9} & +  5.00 y_{10} & +  1.00 y_{11} & -8.00 y_{12} & +  5.00 y_{13} & -3.00 y_{14} & -5.00 y_{15}\\
 y_{5}   &  1.0 & +  4.00 y_{9} & +  8.00 y_{10} & +  6.00 y_{11} & +  3.00 y_{12} & -5.00 y_{13} & -8.00 y_{14} &   \\
 y_{6}   &  1.0 & +  3.00 y_{9} & +  1.00 y_{10} & +  4.00 y_{11} & -6.00 y_{12} & +  2.00 y_{13} & -5.00 y_{14} & +  7.00 y_{15}\\
 y_{7}   &  1.0 & +  7.00 y_{9} & +  3.00 y_{10} & -3.00 y_{11} & -10.00 y_{12} & -1.00 y_{13} & -1.00 y_{14} & -4.00 y_{15}\\
 y_{8}   &  1.0 & -7.00 y_{9} & +  1.00 y_{10} & -5.00 y_{11} & -2.00 y_{12} & -6.00 y_{13} & +  5.00 y_{14} & -8.00 y_{15}\\
\hline
z    &  -0 & +  6.00 y_{9} & -26.00 y_{10} & -18.00 y_{11} & +  5.00 y_{12} &   & -1.00 y_{14} & + 17.00 y_{15}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 11 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{14}   &  0.0291262135922 & -0.08 y_{4} & -0.48 y_{10} & +  1.22 y_{11} & -0.07 y_{3} & +  0.12 y_{8} & +  0.56 y_{9} & +  0.98 y_{13}\\
 y_{12}   &  0.0291262135922 & -0.08 y_{4} & +  1.09 y_{10} & -0.50 y_{11} & +  0.07 y_{3} & -0.02 y_{8} & -0.30 y_{9} & +  0.41 y_{13}\\
 y_{7}   &  0.135922330097 & +  0.97 y_{4} & -5.59 y_{10} & -0.31 y_{11} & -0.35 y_{3} & +  0.24 y_{8} & + 11.26 y_{9} & -5.10 y_{13}\\
 y_{1}   &  1.60194174757 & -0.27 y_{4} & + 10.09 y_{10} & -2.09 y_{11} & +  0.31 y_{3} & -0.64 y_{8} & -11.98 y_{9} & -5.86 y_{13}\\
 y_{5}   &  0.854368932039 & +  0.39 y_{4} & + 15.13 y_{10} & -5.24 y_{11} & +  0.80 y_{3} & -1.05 y_{8} & -1.35 y_{9} & -11.61 y_{13}\\
 y_{6}   &  1.63106796117 & +  0.65 y_{4} & -6.25 y_{10} & +  2.70 y_{11} & -0.48 y_{3} & -0.80 y_{8} & -1.14 y_{9} & -7.02 y_{13}\\
 y_{2}   &  2.28155339806 & -0.42 y_{4} & -11.58 y_{10} & + 26.50 y_{11} & -1.87 y_{3} & +  1.00 y_{8} & -3.10 y_{9} & +  5.37 y_{13}\\
 y_{15}   &  0.135922330097 & -0.03 y_{4} & -0.45 y_{10} & +  0.26 y_{11} & -0.06 y_{3} & -0.04 y_{8} & -0.45 y_{9} & -0.24 y_{13}\\
\hline
z    &  2.42718446602 & -0.81 y_{4} & -27.72 y_{10} & -17.26 y_{11} & -0.67 y_{3} & -0.95 y_{8} & -3.75 y_{9} & -3.02 y_{13}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.805825242718 & +  0.08 x_{14} & +  0.08 x_{12} & -0.97 x_{7} & +  0.27 x_{1} & -0.39 x_{5} & -0.65 x_{6} & +  0.42 x_{2} & +  0.03 x_{15}\\
 x_{10}   &  27.7184466019 & +  0.48 x_{14} & -1.09 x_{12} & +  5.59 x_{7} & -10.09 x_{1} & -15.13 x_{5} & +  6.25 x_{6} & + 11.58 x_{2} & +  0.45 x_{15}\\
 x_{11}   &  17.2621359223 & -1.22 x_{14} & +  0.50 x_{12} & +  0.31 x_{7} & +  2.09 x_{1} & +  5.24 x_{5} & -2.70 x_{6} & -26.50 x_{2} & -0.26 x_{15}\\
 x_{3}   &  0.669902912621 & +  0.07 x_{14} & -0.07 x_{12} & +  0.35 x_{7} & -0.31 x_{1} & -0.80 x_{5} & +  0.48 x_{6} & +  1.87 x_{2} & +  0.06 x_{15}\\
 x_{8}   &  0.95145631068 & -0.12 x_{14} & +  0.02 x_{12} & -0.24 x_{7} & +  0.64 x_{1} & +  1.05 x_{5} & +  0.80 x_{6} & -1.00 x_{2} & +  0.04 x_{15}\\
 x_{9}   &  3.74757281553 & -0.56 x_{14} & +  0.30 x_{12} & -11.26 x_{7} & + 11.98 x_{1} & +  1.35 x_{5} & +  1.14 x_{6} & +  3.10 x_{2} & +  0.45 x_{15}\\
 x_{13}   &  3.01941747573 & -0.98 x_{14} & -0.41 x_{12} & +  5.10 x_{7} & +  5.86 x_{1} & + 11.61 x_{5} & +  7.02 x_{6} & -5.37 x_{2} & +  0.24 x_{15}\\
\hline
z    &  -2.42718446602 & -0.03 x_{14} & -0.03 x_{12} & -0.14 x_{7} & -1.60 x_{1} & -0.85 x_{5} & -1.63 x_{6} & -2.28 x_{2} & -0.14 x_{15}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.805825242718 & +  0.08 x_{14} & +  0.08 x_{12} & -0.97 x_{7} & +  0.27 x_{1} & -0.39 x_{5} & -0.65 x_{6} & +  0.42 x_{2} & +  0.03 x_{15}\\
 x_{10}   &  27.7184466019 & +  0.48 x_{14} & -1.09 x_{12} & +  5.59 x_{7} & -10.09 x_{1} & -15.13 x_{5} & +  6.25 x_{6} & + 11.58 x_{2} & +  0.45 x_{15}\\
 x_{11}   &  17.2621359223 & -1.22 x_{14} & +  0.50 x_{12} & +  0.31 x_{7} & +  2.09 x_{1} & +  5.24 x_{5} & -2.70 x_{6} & -26.50 x_{2} & -0.26 x_{15}\\
 x_{3}   &  0.669902912621 & +  0.07 x_{14} & -0.07 x_{12} & +  0.35 x_{7} & -0.31 x_{1} & -0.80 x_{5} & +  0.48 x_{6} & +  1.87 x_{2} & +  0.06 x_{15}\\
 x_{8}   &  0.95145631068 & -0.12 x_{14} & +  0.02 x_{12} & -0.24 x_{7} & +  0.64 x_{1} & +  1.05 x_{5} & +  0.80 x_{6} & -1.00 x_{2} & +  0.04 x_{15}\\
 x_{9}   &  3.74757281553 & -0.56 x_{14} & +  0.30 x_{12} & -11.26 x_{7} & + 11.98 x_{1} & +  1.35 x_{5} & +  1.14 x_{6} & +  3.10 x_{2} & +  0.45 x_{15}\\
 x_{13}   &  3.01941747573 & -0.98 x_{14} & -0.41 x_{12} & +  5.10 x_{7} & +  5.86 x_{1} & + 11.61 x_{5} & +  7.02 x_{6} & -5.37 x_{2} & +  0.24 x_{15}\\
\hline
z    &  -0.757281553398 & -0.95 x_{14} & -0.10 x_{12} & +  5.21 x_{7} & +  3.09 x_{1} & + 11.34 x_{5} & +  9.85 x_{6} & -4.41 x_{2} & -0.07 x_{15}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.805825242718 & +  0.08 x_{14} & +  0.08 x_{12} & -0.97 x_{7} & +  0.27 x_{1} & -0.39 x_{5} & -0.65 x_{6} & +  0.42 x_{2} & +  0.03 x_{15}\\
 x_{10}   &  27.7184466019 & +  0.48 x_{14} & -1.09 x_{12} & +  5.59 x_{7} & -10.09 x_{1} & -15.13 x_{5} & +  6.25 x_{6} & + 11.58 x_{2} & +  0.45 x_{15}\\
 x_{11}   &  17.2621359223 & -1.22 x_{14} & +  0.50 x_{12} & +  0.31 x_{7} & +  2.09 x_{1} & +  5.24 x_{5} & -2.70 x_{6} & -26.50 x_{2} & -0.26 x_{15}\\
 x_{3}   &  0.669902912621 & +  0.07 x_{14} & -0.07 x_{12} & +  0.35 x_{7} & -0.31 x_{1} & -0.80 x_{5} & +  0.48 x_{6} & +  1.87 x_{2} & +  0.06 x_{15}\\
 x_{8}   &  0.95145631068 & -0.12 x_{14} & +  0.02 x_{12} & -0.24 x_{7} & +  0.64 x_{1} & +  1.05 x_{5} & +  0.80 x_{6} & -1.00 x_{2} & +  0.04 x_{15}\\
 x_{9}   &  3.74757281553 & -0.56 x_{14} & +  0.30 x_{12} & -11.26 x_{7} & + 11.98 x_{1} & +  1.35 x_{5} & +  1.14 x_{6} & +  3.10 x_{2} & +  0.45 x_{15}\\
 x_{13}   &  3.01941747573 & -0.98 x_{14} & -0.41 x_{12} & +  5.10 x_{7} & +  5.86 x_{1} & + 11.61 x_{5} & +  7.02 x_{6} & -5.37 x_{2} & +  0.24 x_{15}\\
\hline
z    &  -0.757281553398 & -0.95 x_{14} & -0.10 x_{12} & +  5.21 x_{7} & +  3.09 x_{1} & + 11.34 x_{5} & +  9.85 x_{6} & -4.41 x_{2} & -0.07 x_{15}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  1.39461883408 & +  0.14 x_{14} & +  0.02 x_{12} & -0.66 x_{7} & -0.88 x_{3} & -1.09 x_{5} & -0.23 x_{6} & +  2.06 x_{2} & +  0.09 x_{15}\\
 x_{10}   &  5.86098654709 & -1.96 x_{14} & +  1.13 x_{12} & -5.81 x_{7} & + 32.63 x_{3} & + 11.07 x_{5} & -9.41 x_{6} & -49.33 x_{2} & -1.63 x_{15}\\
 x_{11}   &  21.7892376682 & -0.71 x_{14} & +  0.04 x_{12} & +  2.67 x_{7} & -6.76 x_{3} & -0.19 x_{5} & +  0.54 x_{6} & -13.88 x_{2} & +  0.17 x_{15}\\
 x_{1}   &  2.16591928251 & +  0.24 x_{14} & -0.22 x_{12} & +  1.13 x_{7} & -3.23 x_{3} & -2.60 x_{5} & +  1.55 x_{6} & +  6.04 x_{2} & +  0.21 x_{15}\\
 x_{8}   &  2.33632286996 & +  0.03 x_{14} & -0.12 x_{12} & +  0.48 x_{7} & -2.07 x_{3} & -0.61 x_{5} & +  1.79 x_{6} & +  2.86 x_{2} & +  0.17 x_{15}\\
 x_{9}   &  29.6995515695 & +  2.35 x_{14} & -2.33 x_{12} & +  2.28 x_{7} & -38.74 x_{3} & -29.76 x_{5} & + 19.73 x_{6} & + 75.42 x_{2} & +  2.92 x_{15}\\
 x_{13}   &  15.7085201794 & +  0.44 x_{14} & -1.70 x_{12} & + 11.72 x_{7} & -18.94 x_{3} & -3.60 x_{5} & + 16.11 x_{6} & + 29.99 x_{2} & +  1.45 x_{15}\\
\hline
z    &  5.93273542601 & -0.21 x_{14} & -0.78 x_{12} & +  8.70 x_{7} & -9.99 x_{3} & +  3.32 x_{5} & + 14.64 x_{6} & + 14.23 x_{2} & +  0.57 x_{15}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  1.63918181818 & +  0.06 x_{14} & +  0.07 x_{12} & -0.91 x_{7} & +  0.48 x_{3} & -0.63 x_{5} & -0.62 x_{6} & -0.04 x_{10} & +  0.02 x_{15}\\
 x_{2}   &  0.118818181818 & -0.04 x_{14} & +  0.02 x_{12} & -0.12 x_{7} & +  0.66 x_{3} & +  0.22 x_{5} & -0.19 x_{6} & -0.02 x_{10} & -0.03 x_{15}\\
 x_{11}   &  20.1396363636 & -0.16 x_{14} & -0.28 x_{12} & +  4.31 x_{7} & -15.94 x_{3} & -3.30 x_{5} & +  3.19 x_{6} & +  0.28 x_{10} & +  0.63 x_{15}\\
 x_{1}   &  2.88309090909 & +  0.00 x_{14} & -0.08 x_{12} & +  0.42 x_{7} & +  0.76 x_{3} & -1.24 x_{5} & +  0.40 x_{6} & -0.12 x_{10} & +  0.01 x_{15}\\
 x_{8}   &  2.67572727273 & -0.08 x_{14} & -0.06 x_{12} & +  0.14 x_{7} & -0.18 x_{3} & +  0.03 x_{5} & +  1.25 x_{6} & -0.06 x_{10} & +  0.08 x_{15}\\
 x_{9}   &  38.661 & -0.65 x_{14} & -0.60 x_{12} & -6.61 x_{7} & + 11.15 x_{3} & -12.84 x_{5} & +  5.35 x_{6} & -1.53 x_{10} & +  0.43 x_{15}\\
 x_{13}   &  19.272 & -0.75 x_{14} & -1.01 x_{12} & +  8.18 x_{7} & +  0.90 x_{3} & +  3.13 x_{5} & + 10.39 x_{6} & -0.61 x_{10} & +  0.46 x_{15}\\
\hline
z    &  7.62336363636 & -0.77 x_{14} & -0.45 x_{12} & +  7.03 x_{7} & -0.57 x_{3} & +  6.52 x_{5} & + 11.93 x_{6} & -0.29 x_{10} & +  0.09 x_{15}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{1} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.171472482436 & +  0.06 x_{14} & +  0.11 x_{12} & -1.12 x_{7} & +  0.10 x_{3} & +  0.51 x_{1} & -0.83 x_{6} & +  0.02 x_{10} & +  0.01 x_{15}\\
 x_{2}   &  0.63956381733 & -0.04 x_{14} & +  0.01 x_{12} & -0.04 x_{7} & +  0.80 x_{3} & -0.18 x_{1} & -0.12 x_{6} & -0.04 x_{10} & -0.03 x_{15}\\
 x_{11}   &  12.4727751756 & -0.17 x_{14} & -0.07 x_{12} & +  3.19 x_{7} & -17.96 x_{3} & +  2.66 x_{1} & +  2.13 x_{6} & +  0.61 x_{10} & +  0.61 x_{15}\\
 x_{5}   &  2.32098946136 & +  0.00 x_{14} & -0.07 x_{12} & +  0.34 x_{7} & +  0.61 x_{3} & -0.81 x_{1} & +  0.32 x_{6} & -0.10 x_{10} & +  0.01 x_{15}\\
 x_{8}   &  2.73733899297 & -0.08 x_{14} & -0.06 x_{12} & +  0.15 x_{7} & -0.16 x_{3} & -0.02 x_{1} & +  1.26 x_{6} & -0.06 x_{10} & +  0.08 x_{15}\\
 x_{9}   &  8.868779274 & -0.68 x_{14} & +  0.24 x_{12} & -10.94 x_{7} & +  3.30 x_{3} & + 10.33 x_{1} & +  1.21 x_{6} & -0.26 x_{10} & +  0.36 x_{15}\\
 x_{13}   &  26.5320550351 & -0.75 x_{14} & -1.21 x_{12} & +  9.24 x_{7} & +  2.81 x_{3} & -2.52 x_{1} & + 11.40 x_{6} & -0.92 x_{10} & +  0.47 x_{15}\\
\hline
z    &  22.7452429742 & -0.76 x_{14} & -0.88 x_{12} & +  9.22 x_{7} & +  3.41 x_{3} & -5.25 x_{1} & + 14.03 x_{6} & -0.93 x_{10} & +  0.13 x_{15}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{11} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.238154420394 & +  0.06 x_{14} & +  0.11 x_{12} & -1.10 x_{7} & -0.01 x_{11} & +  0.52 x_{1} & -0.81 x_{6} & +  0.02 x_{10} & +  0.02 x_{15}\\
 x_{2}   &  1.19416237449 & -0.05 x_{14} & +  0.01 x_{12} & +  0.10 x_{7} & -0.04 x_{11} & -0.06 x_{1} & -0.02 x_{6} & -0.02 x_{10} & -0.00 x_{15}\\
 x_{3}   &  0.694467988004 & -0.01 x_{14} & -0.00 x_{12} & +  0.18 x_{7} & -0.06 x_{11} & +  0.15 x_{1} & +  0.12 x_{6} & +  0.03 x_{10} & +  0.03 x_{15}\\
 x_{5}   &  2.74547692007 & -0.00 x_{14} & -0.07 x_{12} & +  0.45 x_{7} & -0.03 x_{11} & -0.71 x_{1} & +  0.39 x_{6} & -0.08 x_{10} & +  0.03 x_{15}\\
 x_{8}   &  2.62511817056 & -0.08 x_{14} & -0.06 x_{12} & +  0.12 x_{7} & +  0.01 x_{11} & -0.05 x_{1} & +  1.24 x_{6} & -0.07 x_{10} & +  0.07 x_{15}\\
 x_{9}   &  11.1619873843 & -0.71 x_{14} & +  0.23 x_{12} & -10.35 x_{7} & -0.18 x_{11} & + 10.82 x_{1} & +  1.60 x_{6} & -0.15 x_{10} & +  0.47 x_{15}\\
 x_{13}   &  28.4820951232 & -0.77 x_{14} & -1.22 x_{12} & +  9.74 x_{7} & -0.16 x_{11} & -2.10 x_{1} & + 11.73 x_{6} & -0.82 x_{10} & +  0.57 x_{15}\\
\hline
z    &  25.1116385774 & -0.79 x_{14} & -0.89 x_{12} & +  9.83 x_{7} & -0.19 x_{11} & -4.74 x_{1} & + 14.43 x_{6} & -0.82 x_{10} & +  0.24 x_{15}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  0.292672789368 & +  0.07 x_{14} & +  0.13 x_{12} & -1.35 x_{7} & -0.01 x_{11} & +  0.64 x_{1} & -1.23 x_{4} & +  0.03 x_{10} & +  0.02 x_{15}\\
 x_{2}   &  1.1871713722 & -0.05 x_{14} & +  0.00 x_{12} & +  0.13 x_{7} & -0.04 x_{11} & -0.08 x_{1} & +  0.03 x_{4} & -0.02 x_{10} & -0.01 x_{15}\\
 x_{3}   &  0.729110539125 & -0.00 x_{14} & +  0.01 x_{12} & +  0.02 x_{7} & -0.06 x_{11} & +  0.22 x_{1} & -0.15 x_{4} & +  0.04 x_{10} & +  0.04 x_{15}\\
 x_{5}   &  2.86098230292 & +  0.03 x_{14} & -0.02 x_{12} & -0.09 x_{7} & -0.04 x_{11} & -0.46 x_{1} & -0.49 x_{4} & -0.07 x_{10} & +  0.03 x_{15}\\
 x_{8}   &  2.9875459453 & +  0.01 x_{14} & +  0.10 x_{12} & -1.55 x_{7} & +  0.00 x_{11} & +  0.75 x_{1} & -1.52 x_{4} & -0.03 x_{10} & +  0.10 x_{15}\\
 x_{9}   &  11.6301841818 & -0.59 x_{14} & +  0.43 x_{12} & -12.52 x_{7} & -0.19 x_{11} & + 11.85 x_{1} & -1.97 x_{4} & -0.11 x_{10} & +  0.51 x_{15}\\
 x_{13}   &  31.9158913137 & +  0.09 x_{14} & +  0.31 x_{12} & -6.16 x_{7} & -0.23 x_{11} & +  5.44 x_{1} & -14.42 x_{4} & -0.48 x_{10} & +  0.81 x_{15}\\
\hline
z    &  29.3351127225 & +  0.27 x_{14} & +  0.99 x_{12} & -9.72 x_{7} & -0.28 x_{11} & +  4.54 x_{1} & -17.73 x_{4} & -0.39 x_{10} & +  0.55 x_{15}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{5} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  4.28597048019 & +  0.11 x_{14} & +  0.11 x_{12} & -1.48 x_{7} & -0.06 x_{11} & -1.40 x_{5} & -1.91 x_{4} & -0.06 x_{10} & +  0.07 x_{15}\\
 x_{2}   &  0.704443188488 & -0.05 x_{14} & +  0.00 x_{12} & +  0.15 x_{7} & -0.04 x_{11} & +  0.17 x_{5} & +  0.11 x_{4} & -0.00 x_{10} & -0.01 x_{15}\\
 x_{3}   &  2.12119861748 & +  0.01 x_{14} & +  0.00 x_{12} & -0.03 x_{7} & -0.07 x_{11} & -0.49 x_{5} & -0.38 x_{4} & +  0.00 x_{10} & +  0.05 x_{15}\\
 x_{1}   &  6.20960589527 & +  0.05 x_{14} & -0.04 x_{12} & -0.19 x_{7} & -0.08 x_{11} & -2.17 x_{5} & -1.05 x_{4} & -0.14 x_{10} & +  0.07 x_{15}\\
 x_{8}   &  7.65132708736 & +  0.05 x_{14} & +  0.08 x_{12} & -1.70 x_{7} & -0.06 x_{11} & -1.63 x_{5} & -2.31 x_{4} & -0.14 x_{10} & +  0.16 x_{15}\\
 x_{9}   &  85.2209203748 & +  0.06 x_{14} & +  0.02 x_{12} & -14.80 x_{7} & -1.14 x_{11} & -25.72 x_{5} & -14.44 x_{4} & -1.81 x_{10} & +  1.39 x_{15}\\
 x_{13}   &  65.7123991914 & +  0.38 x_{14} & +  0.12 x_{12} & -7.20 x_{7} & -0.67 x_{11} & -11.81 x_{5} & -20.15 x_{4} & -1.26 x_{10} & +  1.22 x_{15}\\
\hline
z    &  57.5245853531 & +  0.51 x_{14} & +  0.83 x_{12} & -10.59 x_{7} & -0.65 x_{11} & -9.85 x_{5} & -22.51 x_{4} & -1.05 x_{10} & +  0.89 x_{15}\\
\end{array}\]


 $ x_{12} $ enters and $ x_{1} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  23.2086662569 & +  0.27 x_{14} & -3.05 x_{1} & -2.07 x_{7} & -0.30 x_{11} & -8.01 x_{5} & -5.11 x_{4} & -0.50 x_{10} & +  0.30 x_{15}\\
 x_{2}   &  1.56699446835 & -0.05 x_{14} & -0.14 x_{1} & +  0.12 x_{7} & -0.05 x_{11} & -0.13 x_{5} & -0.04 x_{4} & -0.02 x_{10} & -0.00 x_{15}\\
 x_{3}   &  2.80055316533 & +  0.02 x_{14} & -0.11 x_{1} & -0.05 x_{7} & -0.08 x_{11} & -0.72 x_{5} & -0.50 x_{4} & -0.01 x_{10} & +  0.06 x_{15}\\
 x_{12}   &  175.574984634 & +  1.54 x_{14} & -28.27 x_{1} & -5.44 x_{7} & -2.25 x_{11} & -61.37 x_{5} & -29.76 x_{4} & -4.07 x_{10} & +  2.11 x_{15}\\
 x_{8}   &  21.3948985864 & +  0.17 x_{14} & -2.21 x_{1} & -2.12 x_{7} & -0.23 x_{11} & -6.43 x_{5} & -4.64 x_{4} & -0.46 x_{10} & +  0.32 x_{15}\\
 x_{9}   &  87.9078057775 & +  0.08 x_{14} & -0.43 x_{1} & -14.88 x_{7} & -1.17 x_{11} & -26.66 x_{5} & -14.90 x_{4} & -1.87 x_{10} & +  1.42 x_{15}\\
 x_{13}   &  85.9861708666 & +  0.56 x_{14} & -3.26 x_{1} & -7.83 x_{7} & -0.93 x_{11} & -18.90 x_{5} & -23.58 x_{4} & -1.73 x_{10} & +  1.46 x_{15}\\
\hline
z    &  203.856791641 & +  1.80 x_{14} & -23.57 x_{1} & -15.13 x_{7} & -2.52 x_{11} & -61.00 x_{5} & -47.32 x_{4} & -4.44 x_{10} & +  2.64 x_{15}\\
\end{array}\]


 $ x_{14} $ enters and $ x_{2} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  32.7278911565 & -6.07 x_{2} & -3.89 x_{1} & -1.33 x_{7} & -0.60 x_{11} & -8.82 x_{5} & -5.33 x_{4} & -0.65 x_{10} & +  0.29 x_{15}\\
 x_{14}   &  34.6870748299 & -22.14 x_{2} & -3.07 x_{1} & +  2.67 x_{7} & -1.09 x_{11} & -2.94 x_{5} & -0.78 x_{4} & -0.55 x_{10} & -0.01 x_{15}\\
 x_{3}   &  3.40816326531 & -0.39 x_{2} & -0.16 x_{1} & +  0.00 x_{7} & -0.10 x_{11} & -0.78 x_{5} & -0.51 x_{4} & -0.02 x_{10} & +  0.06 x_{15}\\
 x_{12}   &  229.034013605 & -34.12 x_{2} & -33.01 x_{1} & -1.33 x_{7} & -3.93 x_{11} & -65.90 x_{5} & -30.96 x_{4} & -4.92 x_{10} & +  2.09 x_{15}\\
 x_{8}   &  27.3537414966 & -3.80 x_{2} & -2.74 x_{1} & -1.67 x_{7} & -0.42 x_{11} & -6.94 x_{5} & -4.78 x_{4} & -0.55 x_{10} & +  0.32 x_{15}\\
 x_{9}   &  90.7006802721 & -1.78 x_{2} & -0.68 x_{1} & -14.67 x_{7} & -1.26 x_{11} & -26.90 x_{5} & -14.96 x_{4} & -1.92 x_{10} & +  1.42 x_{15}\\
 x_{13}   &  105.482993197 & -12.44 x_{2} & -4.99 x_{1} & -6.33 x_{7} & -1.54 x_{11} & -20.55 x_{5} & -24.02 x_{4} & -2.04 x_{10} & +  1.46 x_{15}\\
\hline
z    &  266.238095238 & -39.81 x_{2} & -29.10 x_{1} & -10.33 x_{7} & -4.48 x_{11} & -66.29 x_{5} & -48.71 x_{4} & -5.43 x_{10} & +  2.62 x_{15}\\
\end{array}\]


 $ x_{15} $ enters and $ x_{14} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  778.5 & -482.00 x_{2} & -70.00 x_{1} & + 56.00 x_{7} & -24.00 x_{11} & -72.00 x_{5} & -22.00 x_{4} & -12.50 x_{10} & -21.50 x_{14}\\
 x_{15}   &  2549.5 & -1627.00 x_{2} & -226.00 x_{1} & + 196.00 x_{7} & -80.00 x_{11} & -216.00 x_{5} & -57.00 x_{4} & -40.50 x_{10} & -73.50 x_{14}\\
 x_{3}   &  159.5 & -100.00 x_{2} & -14.00 x_{1} & + 12.00 x_{7} & -5.00 x_{11} & -14.00 x_{5} & -4.00 x_{4} & -2.50 x_{10} & -4.50 x_{14}\\
 x_{12}   &  5553.5 & -3432.00 x_{2} & -505.00 x_{1} & + 408.00 x_{7} & -171.00 x_{11} & -517.00 x_{5} & -150.00 x_{4} & -89.50 x_{10} & -153.50 x_{14}\\
 x_{8}   &  842.5 & -524.00 x_{2} & -75.00 x_{1} & + 61.00 x_{7} & -26.00 x_{11} & -76.00 x_{5} & -23.00 x_{4} & -13.50 x_{10} & -23.50 x_{14}\\
 x_{9}   &  3715.5 & -2315.00 x_{2} & -322.00 x_{1} & + 264.00 x_{7} & -115.00 x_{11} & -334.00 x_{5} & -96.00 x_{4} & -59.50 x_{10} & -104.50 x_{14}\\
 x_{13}   &  3817.0 & -2381.00 x_{2} & -334.00 x_{1} & + 279.00 x_{7} & -118.00 x_{11} & -335.00 x_{5} & -107.00 x_{4} & -61.00 x_{10} & -107.00 x_{14}\\
\hline
z    &  6943.5 & -4301.00 x_{2} & -621.00 x_{1} & + 503.00 x_{7} & -214.00 x_{11} & -632.00 x_{5} & -198.00 x_{4} & -111.50 x_{10} & -192.50 x_{14}\\
\end{array}\]


 $ x_{7} $ enters and Unbounded Dictionary!


 $ x_{7} $ enters and Unbounded Dictionary!
\end{document}
